Learn Russian: Дайте мне меню, пожалуйста.

This is the polite or plural imperative (command) form. You add -те to the basic command form (дай) when speaking to multiple people or respectfully to one person (like a waiter, using the formal "you", вы). If you were asking a close friend to give you something, you would just say дай.

Why is the word for "me" мне here instead of меня?

Мне means "to me" (this is called the dative case). When you ask someone to give, show, or tell something to you, Russian requires this form. Меня is used when you are the direct target of an action, for example, if someone sees you.

Меню is the object being given. Why didn't its ending change?

Меню is a foreign loanword. In Russian, most borrowed words that end in a vowel like -ю, -о (like кино), or -и (like такси) are "indeclinable." This means they never change their endings, no matter what role they play in the sentence.

Is пожалуйста pronounced exactly as it is spelled?

Not quite! It is a bit tricky because you don't pronounce every letter. The й is usually skipped, and the в is completely silent. In everyday speech, it sounds more like "па-жа-лу-ста".